Carbon nanotube array as a van der Waals two-dimensional hyperbolic material

We use an ab initio approach to design and study a two-dimensional material—a planar array of carbon nanotubes separated by an optimal distance defined by the van der Waals interaction. We show that the energy spectrum for an array of quasimetallic nanotubes is described by a strongly anisotropic hyperbolic dispersion and formulate a model low-energy Hamiltonian for its semianalytical treatment. Periodic-potential-induced lifting of the valley degeneracy for an array of zigzag narrow-gap nanotubes leads to the band-gap collapse. In contrast, the band gap is opened in an array of gapless armchair tubes. These unusual spectra, marked by pronounced van Hove singularities in the low-energy density of states, open the opportunity for interesting physical effects and prospective optoelectronic applications.

Figure 1

A horizontally aligned array of $(15,0)$ carbon nanotubes separated by a distance $d$. The diameter of a (15,0) CNT is 22.2 a.u. (1 a.u. = 0.529 Å).

Figure 2

The electron dispersion in a $(15,0)$ carbon nanotube along the tube axis for a small part of the Brillouin zone. The wave vector $k_y$ is given in the reciprocal-space units. Here, $a$ is the translation vector, which is equal to 8 a.u. All the bands are twofold degenerate. There is a small (few meV) curvature-induced band gap at $k_y=0$.

Figure 3

The dependence of the CNT array supercell total energy $V$ in units of the Lennard-Jones potential well depth $\mathrm{\Delta }E$ on the normalized distance between the nanotube walls. A kinky dependence (shown in black) is the result of the PBE-D2 calculations; a smooth (red) curve represents the Lennard-Jones potential. The parameters of the best-fit Lennard-Jones potential are shown in the inset. Here, $\mathrm{\Delta }E$ is the depth of the potential well, and $d=d_0=6$ a.u. corresponds to the potential minimum.

Figure 4

The electronic band structure of an array of $(15,0)$ carbon nanotubes separated by $d=6$ a.u.: (a) along the tube axis for $k_x=0$ and (b) in the direction normal to the tube axis for $k_y=0$. Here, ${\mathrm{\Delta }}_1$ is the band gap between the lowest unoccupied molecular orbital LUMO1 and highest occupied molecular orbital HOMO1 bands in the $\mathrm{\Gamma }$ point; ${\mathrm{\Delta }}_2$ corresponds to the LUMO2 and HOMO2 bands. The wave vectors $k_y$ and $k_x$ are given in reciprocal-space units. The lattice period in the $X$ direction $b=D+d=28.2$ a.u.

Figure 5

The surface plot of the energy dispersion of an array of $(15,0)$ carbon nanotubes separated by $d=6$ a.u. obtained within the PBE-D2 approach.

Figure 6

The electronic band structure of an array of $(9,9)$ carbon nanotubes separated by $d=6$ a.u.: (a) along the tube axis for $k_x=0.2$ and (b) in the direction normal to the tube axis for $k_y=0.33$. The wave vectors $k_x$ and $k_y$ are given in reciprocal-space units. The lattice period in the $X$ direction is $b=D+d=29.07$ a.u. The inset to (a) shows the energy dispersion along the $Y$ axis in the close vicinity of the conduction-band minimum.

Figure 7

The electronic band structure of the optimally spaced ($d=6$ a.u.) array of $(15,0)$ zigzag nanotubes in the direction normal to the tube axis. Solid lines correspond to the band structure described by the effective Hamiltonian; dotted lines show the results of the ab initio calculations. Parameters of the model Hamiltonian are chosen in accordance with Table 2.

Figure 8

The density of states of an array of optimally spaced $(15,0)$ carbon nanotubes calculated using the effective Hamiltonian given by Eq. (2). The sharp peaks correspond to van Hove singularities.

Figure 9

The electronic dispersion of an array of $(12,0)$ carbon nanotubes separated by $d=6.2$ a.u. in the direction normal to the tube axis for $k_y=0$. The lattice period in the $X$ direction $b=D+d=23.96$ a.u.

Figure 10

The electronic dispersion of an array of $(18,0)$ carbon nanotubes separated by $d=6.2$ a.u. in the direction normal to the tube axis for $k_y=0$. The lattice period in the $X$ direction, $b=D+d=32.84$ a.u.